The number of such points (a+1, 3a),  where a is any integer, lying inside the region bounded by the circles x2+y2−2x−3=0  and x2+y2−2x−15=0,  is

The given circles are (x−1)2+y2=4  and (x−1)2+y2=16 The points (a+1, 3a)  lie on the line x=a+1, y=3ai.e. y=3(x−1)  [eliminating a]                whose slope =3,  hence makes angle 600  with the +ve direction of the x−axisA≡(1+2cos600, 2sin600)≡(2, 3)  and B≡(1+4cos600, 4sin600)≡(3, 23)Hence, there is no point on the line segment AB whose abscissa is an integer since abscissa of A is 2 and that of B is 3.